How is a SOF S/S primarily incorporated on the battlefield? For example, is the primary role of a SOF S/S to support the entry team, do SR, or are they sent on "Hunter/Killer" missions (ie: set up on a building in Iraq to destroy targets of opportunity)?
How is their training any different from regular snipers (say Army/USMC S/S)?
Once in your SOF unit, how would you secure a spot to get into the Sniper course to become a sniper (is it primarily through shooting competitions where you have to compete to earn the spot?).

First, apologies for any spelling or grammer errors that turn up in this post, as SP's godforsaken add for The Pretender covers half of the reply box. (Hint for SP sys admin: If I was ever thinking about buying this product, you can be sure I won't do so now.)
That said, different units approach the use of snipers in various ways. In some organizations, you have dedicated sniper teams as part of the reconnaissance capabilities. In other you have some personnel trained as snipers as a sort of additional skill set to be used as needed. In some you have both, either officially on the TO&E or unofficially in how things work out in the real world.
As for how they are incorporated on the battlefield: Yes. Everything you suggested, plus other taskings as necessary, appropriate, etc. Again, you wil see some variation between SOF units (and between SOF units and conventional forces) but good sniper assets look for businness if none is otherwise coming their way.
Differences in training: SOF sniper training, obviously, involves guys who are already in special operations, and so is less focused on "gut check" type stuff ("Ranger School in ghillie suits" as one guy I work with puts it) and more focused on actually training snipers. Probably not a whole lot of difference in course material (+/- training on SOF specific weapons and kit) but different mindset, more of a "gentleman's course" (though graded and with failure entirely possible).
Getting into sniper school once in SOF: I can't speak for the Rangers, SEALs, etc., but for SF it's mostly a question of saying you want to go. It used to be harder to lay hands on a SOTIC slot, you might spend most of your career trying to get a SOTIC Level I slot, but it's different today. First, there's high demand, and second with so many units deployed if you are stateside there are lots of slots to go around, making it much less difficult to lay hands on one. If you're really a non-shooter I would imagine it is much harder but most people who make it into SOF and remain there are proficient enough to handle sniper training.